The weight two \(K\)-theory of fields
Let \(K_p(F, \mathbb{Z}/n)\) denote the \(K\)-groups of a field \(F\), with finite coefficients. For \((n,\text{char } F)= 1\), there are Chern class maps \[ c_{q, p}: \text{gr}^q_\gamma K_{2q- p}(F, \mathbb{Z}/n)\to H^p_{\text{ét}}(F, \mu^{\otimes q}_n); \] it is expected (a version of the Quillen-Lichtenbaum conjectures) that the maps \(c_{q,p}\) should be isomorphisms after inverting ``small'' primes. The author proves (theorem 6.8) that \(c_{2,p}\) are isomorphisms after inverting \((3- p)!\), \(p= 2, 1, 0, -1,\dots\) and for \((n, \text{char } F)= 1\). It implies, when \(F\) is a number field, that \[ \text{gr}^2_\gamma K_p(F)[1/(p- 1)!]= 0\qquad\text{for}\quad p\geq 4 \] and suggests the refined version of the conjecture of Soulé and Beilinson: \[ \text{gr}^q_\gamma K_p(F)[1/(p- 1)!]= 0\qquad\text{for}\quad p\geq 2q. \] Theorem 6.8 is deduced from Hilbert's Theorem 90 for the relative \(K_2\) of regular semi-local rings containing a field, the main technical result of the paper.
scientific article; zbMATH DE number 827887
scientific article